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The Mathematics of Game Shows

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The Mathematics of Game Shows The Mathematics of Game Shows Bowen Kerins Research Scientist, EDC (and part-time game consultant) [email protected] Overview Game shows are filled with math problems… • Contestants • How do I play best? • How much is enough? • Producers • How do I build a fun game to watch? • How will contestants behave? • How much money are we giving out? PRIZES! Want to win? We’ll need some volunteers for games. You may leave here with fabulous prizes! (Warning: definition of fabulous may vary.) Personal Encounters February 2000: Millionaire (episode #49) (for $1000: How many degrees in a right angle?) Personal Encounters February 2000: Millionaire (episode #49) (I got the next question wrong.) Personal Encounters April 2004: The Price Is Right (Double overbid on the showcase! Bummer.) Personal Encounters July 2007: National Bingo Night (I also worked on “Show Me The Money” and “The Singing Bee”… which, four years later, is still on the air.) Personal Encounters Since Then… Recent work: “in-development” shows for Endemol, Gurinco, and Ryan Seacrest Productions. (He’s older than me.) Expected Value Hour $.01 $1,000 $1 $5,000 $5 $10,000 $10 $25,000 $25 Better known as $50,000 $50 $75,000 $75 Deal or No Deal $100,000 $100 $200,000 $200 $300,000 $300 $400,000 $400 $500,000 $500 $750,000 $750 $1,000,000 Expected Value Hour $.01 The “fair deal”: $1,000 $1 $5,000 $5 $10,000 $10 Multiply each $25,000 $25 outcome by its $50,000 $50 probability… $75,000 $75 $100,000 $100 $200,000 $200 Total: $410,210 $300,000 $300 $400,000 $400 $500,000 $500 $750,000 $750 Fair deal: ~$102,500 $1,000,000 Expected Value Hour $.01 The “bank offer”: $1,000 $1 $5,000 $5 $10,000 $10 Guarantee, almost $25,000 $25 always less than fair $50,000 $50 value $75,000 $75 $100,000 $100 $200,000 $200 Fair deal: ~$102,500 $300,000 $300 $400,000 $400 Offer: $82,000 $500,000 $500 $750,000 $750 Deal or No Deal? $1,000,000 Expected Value Hour $.01 What’s the expected $1,000 $1 value of the initial $5,000 $5 $10,000 $10 board? $25,000 $25 $50,000 $50 How does it compare $75,000 $75 $100,000 $100 to the first offer? $200,000 $200 $300,000 $300 How does it compare $400,000 $400 to how much money $500,000 $500 players actually win? $750,000 $750 $1,000,000 Expected Value Hour $.01 $1,000 $1 Initial board… $5,000 $5 Fair deal: $131,477 $10,000 $10 $25,000 $25 $50,000 $50 First offer: $75,000 $75 $100,000 $100 ~$8,000-$20,000 $200,000 $200 $300,000 $300 $400,000 $400 The first offers are $500,000 $500 terrible! Why? $750,000 $750 $1,000,000 Expected Value Hour $.01 $1,000 $1 Actual average $5,000 $5 winnings per player: $10,000 $10 $25,000 $25 $122,500 $50,000 $50 $75,000 $75 Initial board’s $100,000 $100 $200,000 expected value: $200 $300,000 $300 $131,477 $400,000 $400 $500,000 $500 $750,000 $750 (Close! Why the difference?) $1,000,000 Expected Value Hour $.01 $1,000 $1 Offer percentages $5,000 $5 (compared to fair $10,000 $10 value, by round): $25,000 $25 $50,000 $50 11%, 21%, 36%, $75,000 $75 50%, 62%, 73%, $100,000 $100 88%, 92%, 98% $200,000 $200 $300,000 $300 $400,000 $400 $500,000 $500 (Commercial break…) $750,000 $750 $1,000,000 Sponsored by… CME Project • Four-year, NSF-funded curriculum written by EDC • Published in 2008 by Pearson Education • 25,000+ students use it nationally: Boston, Chicago, Pittsburgh, Des Moines… and more Fundamental Organizing Principle The widespread utility and efectiveness of mathematics come not just from mastering specific skills, topics, and techniques, but more importantly, from developing the ways of thinking—the habits of mind—used to create the results. CME Project Overview By focusing on habits of mind… • Coherent curriculum, fewer chapters • Closely aligned to Common Core’s Standards of Mathematical Practice (several ideas come from CME) • Closely aligned to NCTM’s Reasoning and Sense- Making goals (several examples come from CME) • General-purpose tools help students get the big ideas Summer sessions in New England! June 27-29, July 18-20, August 1-5 (we also do house calls… but now, back to the show) The Price Is Right • Now in its 39th year • Lots of good math problems! • Also a huge sample size of repeatedly- played games (for agonizing detail, visit http://tpirsummaries.8m.com) Who wants to play?? Four Price Tags Place a price next to each item. If it’s the right price, you win the prize! Slumdog Millionaire Glee: The Game Word Wear Jenga Four Price Tags Place a price next to each item. If it’s the right price, you win the prize! Slumdog Millionaire Good luck… you’ll need it. $9.99 Glee: The Game $10.00 Word Wear $10.09 $10.59 Jenga Audience, any advice? Four Price Tags So, how did you do…? Slumdog Millionaire $10.00 Glee: The Game $10.59 Word Wear $9.99 Jenga $10.09 The Producers’ Question If I keep offering this game repeatedly, how many prizes will I give away, on average? Answer by calculating the expected value for the number of prizes per game. There are 24 different ways the player can place the price tags. (Why?) 24 Ain’t That Many Here are the 24 ways to place the price tags: ABCD BACD CABD DABC ABDC BADC CADB DACB ACBD BCAD CBAD DBAC ACDB BCDA CBDA DBCA ADBC BDAC CDAB DCAB ADCB BDCA CDBA DCBA Gotta Score ‘Em All For each of the 24 ways, count the number of price tags placed correctly. ABCD 4 BACD 2 CABD 1 DABC 0 ABDC 2 BADC 0 CADB 0 DACB 1 ACBD 2 BCAD 1 CBAD 2 DBAC 1 ACDB 1 BCDA 0 CBDA 1 DBCA 2 ADBC 1 BDAC 0 CDAB 0 DCAB 0 ADCB 2 BDCA 1 CDBA 0 DCBA 0 The expected value is… This frequency chart shows the number of ways to get each result. # prizes # ways 4 1 2 6 1 8 0 9 TOTAL 24 The expected value is… This frequency chart shows the number of ways to get each result. # prizes # ways The total number of prizes is 4 1 4 x 1 + 2 x 6 + 1 x 8 2 6 1 8 … 24 prizes and 24 ways. Divide to 0 9 find… TOTAL 24 Hey, it’s 1! A Second Opinion Reconsider the problem from how Sue sees it. (Sue Sylvester, from Glee.) ABCD BACD CABD DABC ABDC BADC CADB DACB ACBD BCAD CBAD DBAC ACDB BCDA CBDA DBCA ADBC BDAC CDAB DCAB ADCB BDCA CDBA DCBA A Second Opinion Light up all the places where B (the Glee game) is correctly placed. ABCD BACD CABD DABC ABDC BADC CADB DACB ACBD BCAD CBAD DBAC ACDB BCDA CBDA DBCA ADBC BDAC CDAB DCAB ADCB BDCA CDBA DCBA A Second Opinion Glee is won one-fourth (6 out of 24) of the time… and all the prizes are like that. ABCD BACD CABD DABC ABDC BADC CADB DACB ACBD BCAD CBAD DBAC ACDB BCDA CBDA DBCA ADBC BDAC CDAB DCAB ADCB BDCA CDBA DCBA It’s always 1! With four prizes, each is won 1/4 of the time. 4 x (1/4) = 1 With five prizes, each is won 1/5 of the time. With n prizes, each is won 1/n of the time. n x (1/n) = 1 Clever methods can “beat” enumeration. Two Extensions 1. As the number of prizes grows, what happens to the probability of winning nothing at all? 2. The mean (average) number of prizes given away is always 1. What happens to the standard deviation? These two problems have great conclusions, which this slide is too small to contain… (We’ll be right back.) Sponsored by… Rice-A-Roni • It’s “The San Francisco Treat”! • A favorite since 1958 • 25,000+ people eat it nationally: Boston, Chicago, Pittsburgh, Des Moines… and more (And we’re back.) The #1 Game on TPIR is… PLINKO! The #1 Game on TPIR is… PLINKO! Plinko is played so often that great data is available: 2000-2011: played 308 times! Total chips: 1,227 Total chips in $10,000 space: 176 (14.3%) Total winnings: $2,214,600 ($1,805 per chip) Average winnings per play: $7,190 Backtracking Plinko How much is this Plinko chip worth right now? Each entry is the value Plinko of a chip at that spot. We know the last row… Backtracking 5000 100 500 1000 0 10000 0 1000 500 100 Work from the bottom Plinko up… each number is the mean of the two below it! Backtracking 300 750 500 5000 5000 500 750 300 100 500 1000 0 10000 0 1000 500 100 Work from the bottom Plinko up… each number is the mean of the two below it! 300 525 625 2750 5000 2750 625 525 300 Backtracking 300 750 500 5000 5000 500 750 300 100 500 1000 0 10000 0 1000 500 100 Work from the bottom Plinko up… each number is the mean of the two 413 575 1688 3875 3875 1688 575 413 below it! 300 525 625 2750 5000 2750 625 525 300 Backtracking 300 750 500 5000 5000 500 750 300 100 500 1000 0 10000 0 1000 500 100 Work from the bottom Plinko up… each number is the mean 413 494 1131 2781 3875 2781 1131 494 413 of the two 413 575 1688 3875 3875 1688 575 413 below it! 300 525 625 2750 5000 2750 625 525 300 Backtracking 300 750 500 5000 5000 500 750 300 100 500 1000 0 10000 0 1000 500 100 Work from the bottom Plinko up… each number is 453 813 1956 3328 3328 1956 813 453 the mean 413 494 1131 2781 3875 2781 1131 494 413 of the two 413 575 1688 3875 3875 1688 575 413 below it! 300 525 625 2750 5000 2750 625 525 300 Backtracking 300 750 500 5000 5000 500 750 300 100 500 1000 0 10000 0 1000 500 100 In the long run, it all Plinko evens out.
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